Therefore, a researcher gets the idea to poll various groups and ask them to self-identify if they are a NOVID, and we would test the null hypothesis
H_0:p=.05 versus the alternate H_A:p≠.05.
A researcher intends to poll 2000 people and see what percentage of them self-report being a NOVID.
a.  Use the formula
p±1.96 SE,  where SE=√((p(1-p))/n) to find a range of proportions, where if the null hypothesis is true, we would expect to find this proportion with probability 95%.


what should i study to understand this concept better?

To solve this problem, we need to calculate the standard error (SE) and then find the range of proportions based on the given formula. Here’s a step-by-step explanation:

### Given:
- Null hypothesis: \( H_0: p = 0.05 \)
- Sample size: \( n = 2000 \)
- Confidence level: 95%, which corresponds to \( 1.96 \) standard deviations from the mean in a normal distribution.

### Step 1: Calculate the Standard Error (SE)

The formula for the standard error when estimating a proportion is:
\[
SE = \sqrt{\frac{p(1-p)}{n}}
\]

Substitute \( p = 0.05 \) and \( n = 2000 \):
\[
SE = \sqrt{\frac{0.05(1-0.05)}{2000}}
\]
\[
SE = \sqrt{\frac{0.05 \times 0.95}{2000}}
\]
\[
SE = \sqrt{\frac{0.0475}{2000}}
\]
\[
SE \approx \sqrt{0.00002375} \approx 0.00487
\]

### Step 2: Calculate the 95% Confidence Interval

Now use the formula \( p \pm 1.96 \times SE \):
\[
0.05 \pm 1.96 \times 0.00487
\]
\[
0.05 \pm 0.00954
\]

This gives us the range:
\[
0.04046 \text{ to } 0.05954
\]

Therefore, if the null hypothesis is true, we would expect to find the proportion of NOVIDs in the range of approximately 4.05% to 5.95% with a probability of 95%.

### What to Study to Understand This Concept Better:
1. **Basic Probability Theory**: Understanding probability distributions, particularly the normal distribution.
2. **Statistical Hypothesis Testing**: Learning about null and alternative hypotheses, Type I and Type II errors.
3. **Confidence Intervals**: Understanding the concept of confidence intervals and their interpretation in statistical analysis.
4. **Sampling and Standard Error**: Learning how the standard error is calculated and why it is used in statistical estimation.
5. **Z-scores and the Standard Normal Distribution**: Knowing how to use z-scores to determine probabilities and critical values in hypothesis testing.

### Additional Questions:
1. What would the confidence interval look like for a different sample size, such as 1000?
2. How does increasing the confidence level to 99% affect the confidence interval range?
3. What happens to the standard error if the proportion \( p \) is closer to 0.5?
4. Why do we use \( 1.96 \) as the multiplier for a 95% confidence interval?
5. How would we interpret the results if the observed proportion is outside the calculated confidence interval?

### Tip:
When dealing with proportions, the larger the sample size, the smaller the standard error, leading to a narrower confidence interval for the same confidence level.

Therefore, a researcher gets the idea to poll various groups and ask them to self-identify if they are a NOVID, and we would test the null hypothesis H_0: p = 0.05 versus the alternate H_A: p ≠ 0.05. A researcher intends to poll 2000 people and see what percentage of them self-report being a NOVID. a. Use the formula p ± 1.96 SE, where SE = √((p(1-p))/n) to find a range of proportions, where if the null hypothesis is true, we would expect to find this proportion with probability 95%. What should I study to understand this concept better?

Learn how to test the null hypothesis for proportions, calculate the confidence interval, and understand the key statistical concepts involved. This problem includes step-by-step calculations for a NOVID poll of 2000 people, using the formula p ± 1.96 SE and covering key concepts like standard error, normal distribution, and confidence intervals. Suitable for college-level statistics students.

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